Solvable groups and modular representation theory

Abstract

In [4] the representation theory of finite solvable groups was studied, and under the assumption of solvability, it was shown that several conjectures of R. Brauer arising from modular representation theory were true. These conjectures are presumably true without the assumption of solvability. In this paper I should like to describe some properties of modular representations of solvable groups which are not shared in common by all finite groups. Solvable groups can be characterized by the existence of Sylow p-complements and by the special structure of their principal series; both these features will be exploited. Since one rational prime number p will be fixed for modular representation theory, we shall consider the more general class of p-solvable groups, where a group is p-solvable if it has a composition series all of whose composition factor groups are either p-groups or p'-groups. The main results concern the principal indecomposable representations (the indecomposable projective modules in the language of modules). Suppose (D is a group of order g=pag0, where (p, go) = 1. Let Q be a normal algebraic number field containing the gth roots of unity, and , a fixed prime ideal divisor of p. The residue class field Q* determined by p is then large enough to write the principal absolutely indecomposable representations of (M. If U is such a representation, it is well-known that pa divides the degree u of U. For a p-solvable group, we shall see that u = pav, where (p, v) -1 and v is the p'-part of the degree f of the unique irreducible quotient representation j of U. Moreover, the representation U is the induced representation of (M by a suitable irreducible representation from a Sylow p-complement & of 5. One consequence of this is that algebraic conjugates of principal indecomposable characters of 5 (these are complex-valued functions) are again principal indecomposable characters. The proofs of the above results are based on the reduction methods of [4]. In ?1, where the reduction will be briefly described, we shall draw some facts left unformulated in [4]. These are in the nature of relations between the block and group structures of a p-solvable group. For example, a question of N. Ito's [7] as to necessary and sufficient conditions when all blocks of a solvable group have full defect is answered. ?2 contains the results on the principal indecomposable representations.